# Primes of the form $2n^2 + 2n + 1$

Is it already known that there are infinitely many primes of the form $2n^2 + 2n + 1$? I was searching online for any articles about it but I can't find any so I suppose this is still unknown. I found that there are infinitely many such primes and one result is that there are infinitely many primes of the form $4t_n + 1$ where $t_n$ are triangular numbers.

• It might be worth noting that $2n^2+2n+1=n^2+(n+1)^2$ is the sum of two consecutive perfect squares. – barak manos Nov 7 '16 at 8:24
• Yes indeed that's how I actually found it first. – unknownMe Nov 7 '16 at 8:25
• This paper claims that no polynomial of degree 2 or higher is known to represent infinitely many primes: math.byu.edu/~lzhao/Presentations/primequadprog1.pdf But not for a lack of 94 years of trying. So your result might be met with skepticism. Would you consider posting a clear argument? – alex.jordan Nov 7 '16 at 8:26
• – punctured dusk Nov 7 '16 at 8:28
• ok , yeah i got it – unknownMe Nov 7 '16 at 8:31

Definitely out-of-reach. $$p=2x^2+2x+1\quad\Longleftrightarrow\quad 2p-1 = (2x-1)^2$$ hence you claim is equivalent to
There are infinite values of $n\in\mathbb{N}$ such that $n^2+1$ is twice a prime
There are infinite values of $n\in\mathbb{N}$ such that $n^2+1$ is a prime
is still a conjecture, namely Landau's conjecture. The closest theorem we have at the moment is due to Iwaniec and Friedlander: there are an infinite number of primes of the form $a^2+b^4$.